Symmetries of Excitons: Implications for Exciton-Phonon Coupling and Optical Spectroscopy

Excitons, bound electron-hole pairs, play a crucial role in governing light-matter interactions in two-dimensional materials and wide-bandgap insulators such as hexagonal boron nitride. To obtain the energies and eigenstates of excitons, the state-of-the-art method is to solve the Bethe-Salpeter equation (BSE). Over the past two decades, many new approaches have been developed to compute exciton dynamics, including their coupling with phonons. These methods have been successful in determining properties such as exciton lifetimes and in understanding the optical spectra associated with them. Despite this progress, the symmetries of excitons and the selection rules associated with them have been largely overlooked.

In this thesis, we demonstrate how excitonic states transform under the action of crystal symmetry operations. In particular, we present a robust method for computing the representations of excitonic states. We apply this framework to analyze the selection rules governing exciton-photon and exciton-phonon interactions, which manifest themselves in spectroscopic techniques such as resonant Raman spectroscopy, absorption, and phonon-assisted luminescence across a wide range of materials.

Furthermore, we explore a particularly intriguing phenomenon in two-dimensional heterostructures: interlayer exciton-phonon coupling. This phenomenon arises from the interaction between excitons and phonons across adjacent layers. Although this phenomenon has been experimentally observed in various heterostructures of layered materials, its microscopic origin and underlying selection rules have remained elusive. Using the WSe$_2$@hBN heterostructure as an example, we investigate the origin of interlayer exciton-phonon coupling and its signatures in resonant Raman scattering through first-principles calculations. With the methods developed in this thesis, we elucidate how crystal symmetries play a central role in governing interlayer exciton-phonon scattering processes, which are responsible for the anomalous resonant Raman intensities of the in-plane and out-of-plane $h$BN phonon modes. Moreover, we address the long-standing question regarding the underlying mechanism of this coupling. In particular, we find that the deformation potential induced by the $h$BN phonon interacts with the hybridized hole density of WSe$_2$ excitons near the $h$BN interface, leading to interlayer exciton-phonon coupling.

Finally, we present three computational tools that enhance state-of-the-art exciton-phonon calculations: (i) \texttt{LetzElPhC}, (ii) \texttt{Ydiago}, and (iii) \texttt{PhdScripts}.

\texttt{LetzElPhC} is a code for the calculation of electron-phonon and exciton-phonon coupling matrix elements. The code utilizes full crystal symmetries, which now makes it possible to perform exciton-phonon calculations using symmetries, drastically decreasing the computation time for these calculations. It resolves the long-standing phase issues that arise when expanding the electron-phonon coupling matrix elements from the irreducible Brillouin zone to the full Brillouin zone, which stem from the incompatibility between the electron-phonon matrix elements and the excitonic wavefunctions. The code also computes the electronic representation matrices as a byproduct, which enable us to compute the representations of the excitonic states.

\texttt{Ydiago} is a diagonalization library for the YAMBO code, which significantly accelerates the diagonalization of full or partial BSE matrices, achieving a tenfold improvement in both speed and memory efficiency compared to existing implementations in the Yambo code.

\texttt{PhdScripts} are a set of Python scripts that allow us to compute the irreducible representation labels for the excitonic states, exciton-phonon matrix elements with full crystal symmetries, as well as resonant Raman intensities and phonon-assisted luminescence intensities. Due to the use of symmetries, these scripts enable a more efficient computation of exciton-phonon properties compared to existing implementations.